Effect of a cylindrical refracting interface on ultrasound intensity and the CW Doppler spectrum

The insonation of axisymmetric flow in a tube by an ultrasound beam which is refracted at the tube surface is considered. A three-dimensional (3-D) model for refraction at a cylindrical interface is developed. The incident beam, which is planar but otherwise arbitrary, is refracted at a single acoustic impedance interface between the medium within the tube and the surrounding medium. A ray approximation is used as the ultrasonic wavelength is small compared to the radius of curvature of the tube. A parametric form for the ray paths is obtained and used to derive an expression for the intensity. The intensity depends on the specific position within the tube, as well as the ratio of sound speeds, N, and the Doppler angle thetaD. A diverging (N = 1.2, thetaD = 45 degrees) and a converging (N = 0.8, thetaD = 45 degrees) case are presented in detail. The converging interface results in more extreme intensity distortion. The regions of maximum and minimum intensity over a tube slice are immediately adjacent in both diverging and converging cases. Once the intensity distribution across a tube slice is known, the previously reported volume integral method for the calculation of continuous wave (CW) Doppler spectral power density (spd) functions may be used. The effect of refraction of the incident beam on the spd function is shown for both an infinite beam and a circular beam with radius equal to the tube radius. A curved acoustic impedance interface, as may be encountered in vitro in flow phantoms or in vivo in vascular disease, leads to nonuniform insonation and spectral distortion which depends on transducer orientation.